In this article we study the existence of solutions of a system of partial differential equations of elliptic type, describing the distribution of a biological species “$u$” and the density of a chemical stimulus “$\psi$” in a bounded domain $\Omega$ of ${\mathbb{R}}^N$. The equation for $u$ includes a chemotaxis term with nonlinear flux limitation which depends on the exponent $p>1$.

The equation for $u$ is given by $-div(M\left(x\right)\nabla u)+u=-\chi div(u{|\nabla \psi |}^{p-2}\nabla \psi )+f\left(x\right),$where $\psi$ presents a subcritical production term $u^{\theta }$ and satisfies the equation $-div(M\left(x\right)\nabla \psi )+\psi =u^{\theta }.$The matrix of coefficients, $M$, is a known, symmetric and positive defined with coefficients $m_{ij}\in C^1\left(\overline{\Omega }\right),$ $\chi$ is a given real constant, $f$ is a non-negative function belonging to $L^m\left(\Omega \right)$, $m>max\{1,\frac{N}{2}\}$. The production term exponent, $\theta$, is assumed to be positive and fulfills one of the following constrains $1<p<\frac{N\theta }{N\theta -1},1<N\theta$or $max\{N,p\}<\frac{\theta +1}{\theta }$ for $\theta >0$.

The problem is completed with Dirichlet boundary conditions for $u$ and $\psi$. The main result of the article includes the existence of positive solutions in $H_0^1\left(\Omega \right)\cap L^{\infty }\left(\Omega \right).$

在本文中，我们研究了椭圆型偏微分方程组解的存在性，描述了生物物种的分布“$你$” 和化学刺激的密度 “$\psi$” 在有界域中$\Omega$的${\mathbb{R}}^{ñ}$. 方程为$你$包括具有非线性通量限制的趋化性项，其取决于指数$p>1$.

方程为$你$是（谁）给的$-d\mathrm{一世}v(米\left(X\right)\nabla 你)+你=-\chi d\mathrm{一世}v(你{|\nabla \psi |}^{p-2}\nabla \psi )+F\left(X\right),$在哪里$\psi$提出了一个亚临界生产术语${你}^{\theta }$并满足方程$-d\mathrm{一世}v(米\left(X\right)\nabla \psi )+\psi ={你}^{\theta }.$系数矩阵，$米$, 是一个已知的、对称的和正定义的系数${米}_{\mathrm{一世}j}\in C^1\left(\overline{\Omega }\right),$ $\chi$是给定的实常数，$F$是一个非负函数，属于${\mathrm{大号}}^{米}\left(\Omega \right)$,$米>最大限度\{1,\frac{ñ}{2}\}$. 生产项指数，$\theta$, 被假定为正并满足以下约束之一$1<p<\frac{ñ\theta }{ñ\theta -1},1<ñ\theta$或者$最大限度\{ñ,p\}<\frac{\theta +1}{\theta }$为了$\theta >0$.

问题用狄利克雷边界条件完成$你$和$\psi$. 本文的主要结果包括存在正解$H_0^1\left(\Omega \right)\cap {\mathrm{大号}}^{\infty }\left(\Omega \right).$